Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

Abstract

The stability of the solution to the equation $(\ast )\dot{u}=F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We assume that the Fréchet derivative of $F(t,u)$ is Hölder continuous of order $q>0$ with respect to $u$ for any fixed $t\ge 0$, i.e., $\Vert F_u^{\prime }(t,w)-F_u^{\prime }(t,v)\Vert \le \alpha (t)\Vert v-w{\Vert }^q$, $q>0$. We proved that the equilibrium solution $v=0$ to the equation $\dot{v}=F(t,v)$ is Lyapunov stable under persistently acting perturbation $f(t)$ if ${\sup }_{t\ge 0}{\int }_0^t\alpha (\xi )\Vert U(t,\xi )\Vert d\xi <\infty$ and ${\sup }_{t\ge 0}\Vert U(t)\Vert <\infty$. Here, $U(t):=U(t,0)$ and $U(t,\xi )$ is the solution to the equation $\frac{d}{dt}U(t,\xi )=F_u^{\prime }(t,0)U(t,\xi )$, $t\ge \xi$, $U(\xi ,\xi )=I$, where $I$ is the identity operator in $\mathcal{X}$. Sufficient conditions for the solution $u(t)$ to equation (*) to be bounded and for ${\lim }_{t\to \infty }u(t)=0$ are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.